Tunnel and similar structures



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TUNNEL. AND SIMILAR STRUCTURE Filed Deo. 1. 1939 3 Sheets-Sheet l P19/Oe 427- '765452 50584) 7' r-/Jz/ g (+4001 I F-z'ne of 619123015' F254 l 00402) af' Prassi/res. l

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TUNNEL AND SIMILAR STRUCTURE Filed nechl, 1959 s sheets-sheet 2 PE/oe Aer 5 4.1 3 4 en I0 araolzd rc'h.

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TUNNEL AND SIMILAR STRUCTURE Filed Dec. 1, 1959 :s sheetsf-sheet s 750 aand/9,5, 741.55, .sg nc/z.

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Patented Oct. 27, 1942 UNITED STATES PATENT OFFICE TUNNEL AND SIMILAR STRUCTURES Ernest H. Wilcox, Los Angeles, Calif.

Application December l, 1939, Serial No. 307,075

4 Claims.

This invention has reference to underground tunnels, conduits, etc., for whatever purpose used. Culverts, storm drains, sewers, water conduits, traffic tunnels, are typical. Structures of the type with which the invention is concerned are subjected to external or superposed loads; and the general and primary object of the invention is the provision of a type of construction capable of safely carrying the external loads, whatever they may be and however they may vary, and at the same time to provide the highest eiiciency of construction material and the highest economy in material and construction costs.

My invention has particularly in View structures which are made of concrete, although structures composed of masonry or even of other materials are not excluded from the invention.

Underground conduits and like structures divide themselves into two general classes. First, there is the class which includes conduits like ordinary pressure pipe. In this class of conduit the pipes are usually comparatively small, and although they are subjected to external loads the internal pressures constitute the primary loads and are usually the only ones dealt with in design.

The second general type of buried or underground tunnel or conduit, typied by culverts, drainage conduits, trailic tunnels, etc., is the type in which the external load, including the weight of the structure itself, is usually the major load and the internal pressures are relatively so small as to be neglected. My invention deals primarily with structures of this second mentioned type rather than of the first mentioned type; although, as will be recognised from the following descriptions, my system of design and construction does not exclude the possibility of taking internal loads into consideration in any case Where those loads are large enough to become an appreciable or substantial factor.

The most commonly used forms of the second general class are conduits of circular or oval form, conduits of polygonal, usually rectangular, form, and tunnels or conduits of the form usually described as semi-elliptic. Conduits of the circular form are most commonly used in relatively small sizes. Their use for relatively large sizes and for carrying any substantial amount of external load is highly uneconomical. This fact will be readily recognized from the comparisons that follow between my design and the semielliptic forms which are much more veconomical than the circular. Conduits of the box or rectangular form have been largely used. They are usually but not invariably laid with their longer cross sectional dimension horizontal, but this form also is highly uneconomical where any substantial external load is to be carried. All the faces of such a conduit are exposed to external pressures, so that each Wall must be considered as a beam and reinforced accordingly to take the tensions that `are, developed. And likewise the corners or angles at the wall junctures must be reinforced to transmit to adjacent walls the strains developed in each wall. For such reasons the box form is highly uneconomical if its walls and reinforcements are made heavy enough to achieve safety.

In the past conduits and tunnels of the arch type have been somewhat prevalently used, these structures having a substantially horizontal bottom wall or floor, vertical side walls, and a semicircular arch top. Of more recent years this arch type has been almost universally replaced by the semi-elliptic type which has a horizontal or dished bottom wall or floor, heavy vertical side walls, and an arched upper Wall which, although designed to approximate a semi-ellipse, is usually formed as a series of circular arcs approximating the semi-ellipse. In this form, even though the semi-elliptic top might be made accurately elliptic in shape, it is to be noted that it always includes vertical side walls (which must be comparatively heavy for reasons which will appear hereinafter) and a fully semi-elliptic arch which springs at each side tangentially from the side walls.

As far as I am aware all buried conduit or tunnel structures fall substantially into one or another of the types or classes stated above. All these buried structures are subject to certain stresses which are the resultant of the internal and external pressures; and the structures with which my invention particularly deals are, at certain times and under certain circumstances, subject to relatively heavy, and sometimes Varying, external pressures, and usually subject to none or substantially little of internal pressures. With this type of structure it has usually been the proper and safe practice to consider the external pressures and loads alone, without making any allowances for the counter-loads set up by internal pressures. The external loads which must be considered consist not only of the weight of the superimposed material, usually earth, and the weight of the structure itself, but also the weight of any additional temporary or permanent loads that may be imposed upon the superincumbent material. For instance, in tunnels or conduits buried under streets or roadways, large temporary variations in the total imposed load must be taken into account, and in conduits passing under prospective building sites 1 or the like, permanent imposition of further and sometimes heavy loads must be taken into account. And also, in order to provide a structure which Will be safe and stable under all possible varying conditions, the fluidity or shiftability of the loads or of the superimposed earth must be taken into account. The ideal structure subject to such external loads is one that is so designed that it is in internal stress equilibrium throughout all possible load variations.

Asis well known, such loads as I speak of herein are productive of pressures which are applied to the structure in directions normal to the structure surface at the points of application, and that those pressures produce internal pressures or stresses in the structure; and also as is well known it is necessary to the stability, economy and safety of a concrete, masonry or similar structure that the external pressures do not set up within the structure any negative pressures, or tensions, in any part of the structure; or that if such tensions do occur then a sufficient amount of steel be inserted to take care of the tensions. Among other things, it maybe stated as an object of my invention to provide a design that as far as may be eliminates the possibility of existence of ten- -f sions and therefore eliminates the necessity of using any reenforcement to take the tensions. And it is also an object of my invention, by eliminating tensions and evenly distributing the positive internal forces or compressions, to provide a structure in which wall thickness is minimized. In many cases my invention makes possible and practicable the use of an arch wall whose thickness is limited only by the practicability of construction and not by the stresses to which the wall is subjected.

in any such structure there are ascertainable lines which may be called the lines of centers of internal pressure; which are lines along which all the resultant internal forces applied t0 the structure may be considered as concentrated. If the lineV of center of pressure coincides with the geometric center of any section of the structure, the eccentricity of pressures on that section is zero, and the actual pressures exerted across the section will the-n be evenly distributed with relation to the center. If the line of center of pressure is eccentric with relation to the geometric center of the section then, depending upon the amount of eccentricity, the actual pressures exerted across the section become non-symmetric with relation tothe center of the section. In a section which is symmetric with relation to a center which is also its linear dimension center,

an eccentricity of the line of center of pressure equal to one-sixth of the distance from the geometricV center to the-section edge will cause the pressure at one edge point to fall to zero. And any greater eccentricity will cause the pressure at such edge point to become negative, or tension. It is therefore necessary in such a section to keep the line of center of pressures within the central third of the section in order to preclude the generation of tensions in the section.

The general objectof my invention may thus also be stated to be the provision of such a design for an externally loaded conduit, tunnel or like structure, that the line of center of internal pressure resultant from all applied forces falls always within the center third of all sections of the structure, and preferably well within or at the center of the center third so as to distribute the positive pressure forces as evenly as possible, with the Vresult that `the most economical and safe structureA is produced. And it is also an object of my invention to provide a structure of such design and form that, when subjected to a very large range of external loads, the changing center linev of pressure always falls safely within the center third of the section.

I have discovered, and checked the fact by mathematical and graphic computations and checks, that an arch of partial elliptic form, much less than semi-elliptic, can be made to satisfy all of the necessary conditions as hereinabove stated; and that such a partial elliptic form is unique in that for all ordinarily encountered loads, with all load variations plus and minus that are likely to occur, such a partial elliptic form is the only one which satisfies the conditions. Under extreme conditions, where the external load is so very great that it can be practically regarded as infinite, a parabolic form will satisfy the conditions if the load at no time becomes relatively small. And if the external load is relatively great, the partial elliptic form may approach somewhat closely to parabolic. Under certain conditions then the arch form may be in ideal very nearly parabolic. But under normally encountered conditions, and particularly where there may be substantial lessening of load due to any condition, the partial elliptic form is the preferred and ideal form which my invention contemplates, and is the unique form which only will satisfy the conditions.

In order to show the advantages which are inherent in my invention, in the following description I give some comparisons between it and the form of common use which is closest in apparent form to my designthe so-called semi-elliptic form. Consideration of that commonly used form will be suicient to show the outstanding advantages inherent in my design. It will become apparent to those skilled in the art, without any further specific discussion, that the commonly used circular andbox forms are still more uneconomical in material and construction cost than is the so-called semi-elliptic form.

For the purpose of fully explaining my invention by reference to certain typical and illustrative forms thereof, and for showing by comparison the advantageous features inherent in my invention, I refer to the accompanying drawings in which- Fig. l is a diagrammatic half-section of a standard conduit o-r tunnel arch of the semielliptic form.

Fig. 2 is a similar diagrammatic half-section showing a parabolic form.

Fig. 3 is a similar half-section showing my partial elliptic form.

All three forms shown in the drawings have been designed for substantially identic flow capacities, and the center lines of pressures and the calculated data which are indicated on the drawings have all been calculated for similar loadings, as will be hereinafter explained. The general engineering considerations applied to the design of all the forms are the same, so that in all matters the several forms illustrated are strictly compara-ble.

The arch form shown in Fig. l has been designed in conformity with the best known engineering practice relative to structures of this type. As shown the form includes a bottom or iioor structure A of the usual approved design and a heavy vertical side wall B which extends up to the spring line shown at C. From this spring line to the apex of the arch the inner curved surface D of the arch is in a form closely approximating a semi-ellipse. In the particular design here chosen for purposes of illustration the semi-major axis (vertical) of the equivalent ellipse is 56 inches and the semi-minor axis (horizontal), located at the spring line, is 42 inches. In approximating the elliptic shape, the lower portion of the inner surface curve D is a circle of radius 70 inches and the upper portion a circle of radius 35 inches. However, the curve just described departs from the true ellipse by a maximum of only 0.4 inch, lso that to all intents and purposes, the form of this arch may bev considered as if it were truly of semi-elliptic shape.

The increase in radial dimension of the radial sections indicated in spaced relation along the length of the arch has been adopted from good engineering practice, amounting to an increase, over the radial dimension at the apex, of oneftieth of the horizontal distance of the inner surface edge of the section from the central vertical plane E of the structure. I may say also that in the forms shown in Figs. 2 and 3 this same method of increase of thickness has been used. The sections indicated in Fig. 1, like the corresponding sections indicated in Figs. 2 and 3 and designated -I, I-2, etc., have been located on the arch in suchmanner that their external edges at the outer surface of the arch structure are located at increments of 2 inches in horizontal distance from the central Vertical plane E of the arch. f

The figures of tension and compression which are given on the diagram, and the location of the line of centers of pressures F, have been calculated mathematically and graphically for a load above the crown line equal to 1500 lbs. per sq. ft., a loading which may be taken roughly as being equal to that due to approximately 15 feet of earth over the crown, but is determined as the load equal to 'Z1/2 feet of earth, as dead load, plus an equal allowance for impact due to live loads. This total load above the crown, consisting of the dead load plus the live load, is known and will be referred to as the specified load. The live load is usually figured by adding one-third to one-half to the actual weight of the live load, to allow for impact. The normal load above the crown of course consists only of the dead load, which is the above-crown load at all times except when the live or any other extraordinary load is imposed. The total loading also includes, in addition to the dead or specified load, proportionate loads due to the weight of material bearing on the structure below the crown level, and the weight of the structure itself. The figures which are indicated in parenthesis in connection with the ends of the sections -I, etc., are the figures of resultant compression or tension forces which are exerted on one square inch of section at the indicated points, compressions being indicated with plus signs and tensions with minus signs.

As will be observed, the line F of centers of pressures not only lies far without the center third of the large majority of the indicated sections, but actually lies within the hollow interior of the arch structure itself for a great deal of its arch length. And this line of centers of pressures also lies without the center third of the vertical supporting wall B for the greater part of the height of that wall. As a result it will be noted that on the outer face G of the arch there is a zone of tension which extends from approximately the section denoted 5-6 to a point below the section denoted 25-26. This zone of tension is indicated by the dimensional arrows and the letter T. It will be noted that the tensions run as high as 261 pounds per square inch. It will also be noted that the positive pressures or compressions at the inner face D of the arch vary quite widely and reach a gure around 400 pounds per square inch.

Due to all the above stated facts a considerable quantity of reenforcing steel is necessary in an arch structure of this semi-elliptic type; and also, as will be noted by comparison further along, an excessive amount of concrete is required. A tunnel structure as illustrated, on a gradient or fall of one foot vertical in 100 feet horizontal (s=0.01), has a fiow capacity of 746 cu. ft. per sec. at a velocity a little in excess of 18 ft. per sec., and is estimated to require, per 100 linear feet, something in excess of '77 cu. yds. of concrete and approximately 8000 pounds of reenforcing steel.

Fig. 3 shows a diagrammatic half section of my partial elliptic arch to form a tunnel or conduit of substantially the same flow capacity at the same gradient and at substantially the same Velocity, .as has been chosen for the structure of Fig. 1. For purposes of comparison it may be stated that the comparative capacity of the partial elliptic arch form shown in Fig. 3 is approximately 760 cu. ft. per sec. at a velocity somewhat less than 18 feet per sec. As here shown the structure has a bottom wall or oor structure A similar to that shown in Fig. 1. From this floor structure springs the partial elliptic curve of the inner arch face De. This inner arch face, in the particular illustration here given, is chosen to be the end portion of an ellipse whose semi-major axis (vertical) is 672.5, and whose semi-minor axis (horizontal) is 93.0".

' The vertical height of the inner surface of the crown of the structure is 94" above the lower edges of the elliptic surface where it meets the iioor structure; and the horizontal semi-width at that point is 47.42,.

The arch form shown in Fig. 2 has an inner face curve Dp of parabolic form, laid out to give a tunnel structure of substantially the same ow capacity,lunder the same conditions, as that of the partial elliptic form shown in Fig. 3. The semi-width of this parabodic arch structure at the iioor level is made approximately the same as that of the partial elliptic form, and the parabolic shape is determined by the location of the intersection with the floor structure and the location of the apex.

In both the forms shown in Figs. 2 and 3 the several sections, denoted 0-|, I-Z, etc., are

Y the same basis as for the form shown in Fig. 1.

On the two arch forms shown in Figs. 2 and 3 the limits of the middle third of the sections are indicated by the dotted lines which are so denoted on the figure. For a uniform specified exterior load above the exterior crown of the arch structure equal to 1500 lbs. per sq. ft., plus proportionate loads due to the weight of the material bearing on the structure below the crown level, and adding also the weight of the structure itself, the line of center of pressure for the partial ellipticarch is shown by the full line denoted Fe, and the corresponding line of center of pressure for the parabolic arch is shown by the full line denoted Fp. It will be noted that, for the partial elliptic arch, the center line of pressures Fe lies substantiallyV exactly at the geomet-` ri'ccenters of all the several sections, in fact the variance in the actual structure is at no point as much as two-hundredths of an inch. Y In the parabolic structure it will be noted that the center line Fp lies entirely within the limits ot the middle third, but lies outside the section centers substantially throughout its length, and approaches fairly closely to the outer limit of the middle third in the vicinity f the sections indicated at lil- I4 to l9-2l. Thus, in the partial elliptic orm,`under the specied loading, the unit pressures across all of the' indicated sections are uniformly distributed with reference to the geometric`rv centers ofthe sections. There are no negative or tensile forces at any place in the arch structure, and the positive or compressive forces at all of the sections, 'and at the construction joint shown at J near the footiof the arch, are substantially uniform per sq. in. of section, and are uniformly low. Under the specied loading the highest unit pressure across va section is about 122 pounds per sq. in. (at the lower end of section 25, at the construction joint J) and the lowest is 39 lbs. per sq. in. at the crown.

Comparatively in the parabolic form shown in the left hand half ofFig. 2, under the same specified loading, the unit pressures at the various sections, although always positive, vary from approximately 39 lbs. per sq. in. at the crown to 169 lbs. per sq.,in. at section Ill-2U. If the dead load only be considered thenV the line of center of pressures moves outwardly in the structure. The position of the line of center of pressures for the dead load of 750 lbs. per sq. ft. is shown at Ge in the partial'elliptic arch in Fig. 3 and at Gp in the parabolic arch in Fig. 2. As will be seen in Fig. 3, this position for the pressure center line under the reduced or dead loading is still within the center third of the sections, and consequently, although the positive internal pressures at the several sections are now unevenly distributed, there are'still no negative pressures or tensions developed.V The structure is still in stable condition, withno forces vtending to collapse it.

Comparatively, it will be seen that under the reduced dead loading, the 'center line of pressures Gp in thepar'abolic arch in the left hand half of Fig. 2 isflocated without the center third of the sections throughout the major portion of the arch length, from a 4point in the vicinity of section i' If the .external load on the structure of Fig. 3=

be'increased t0 an extraordinary load above the crown of -3000 lbs. per sq. ft., then the line of center of pressures for that extraordinary load, shown at He in Fig. 3, lies inward of the normal center line Fe,- but is still well within the center third of the sections. And so, under the extraordinary load n0 tensions are developed and the partial elliptic arch is still in equilibrium with no forces tending to push it either inwardly or outwardly. And it will be noted from the relative position of the center line I-Ie, that there is still a wide margin between it and the line delimiting the'inner limit of the center third, so that a considerably greater extraordinary load can still be carried without developing any internal tensions in the structure. And I may further note that, even under the extraordinary load of 3000 lbs. per sq. ft., the maximum unit stress, which occurs in the vicinity of section 23-24a, is only about 275 lbs, per sq. in., which is less than half that conservatively allowable in concrete structures. From what I have said it will now be recognize that the described partialelliptic structure is stable under a wide variety of conditions, which is true of no other type of structure. Being so stable, there is no necessity of utilizing any reenforcing'steel in the arch structure itself. Reenforcing steel will of course be necessary in the 1 floor structure or base to substantially the same extent that it is necessary in any of the other types oi structures, but' is not necessary in the arch structure itself so far as its ultimate loadcarrying ability is concerned. However, it is advisable, to care for temperature changes, and to resist unbalanced loading during the process of back-filling around the structure, to use some reenforcing steel, which, however, may be comparatively light.

Thus, in my partial elliptic structure, the necessity of reenforcing steel is either obviated or reduced. And further, as simple comparison of Figs. 1 and 3 will indicate, the amount of concrete required for the partial elliptic structure is much less than that required for the standard structure of Fig. l. It will be noted that the arch Wall of the partial elliptic structure is much thinner, but at the same time, as I have pointed out, the unit stresses in the partial elliptic structure under any given loading are markedly less than in the standard structure. In fact, in a great many cases the'wall thickness of the partial elliptic structure is limited only to the thinnest section which it is practicable to pour under actual working conditions, rather than limited by the stresses to which the structure will be subjected. One result of all these advantages is that, beside being much more inherently safe, the partial elliptic structure is much less expensive.

Another advantage which my partial elliptic structure entails is in economy with reference to the pouring forms required. In the construction of such things as conduits for water carriage a great many situations may be encountered which are substantially alike as regards superimposed load. Where those conditions are similar, and conduits Vof different capacities are required, it is only-necessary, in design and construction, to lower or raise the floor structure A, so to speak, without having to design and construct an arch to a new elliptic conformation. Thus, if it isdesired to construct a conduct of lesser capacity than that shown in Fig. 3, to sustain substantially the same external load, it is only necessary to make a structure in which the floor line is moved up to the dotted line shown at AI in the figure. And this, as will be recognized, effects very great economy not only in initial calculation and design, but also in the use of forms. If a single set of internal forms be provided, made up in suitable sections, that set of forms may be used without any change other than deleting or adding sections, for the construction of many different sizes of conduit,

For the purpose of enabling those skilled in the art t0 utilize the invention I will outline brieily and broadly a suitable method of arriving at the design required by any set of conditions. This method is merely indicative and not necessarily exclusive. Given any set of conditions, a rst approximation may be laid out, using for that purpose an ellipse which does not differ very greatly from a parabola. A tentative structure design may then be laid out on the assumed ellipse, and the calculations of internal pressures made for that assumed design. I need not go into details relative to a calculation of the irnposed external pressures, including intrinsic Weights of the structure, and the transformation of those pressures to obtain the force moments resultant in the structure, and the following determination of lines of centers of internal pressures, as the appropriate mathematical and graphic methods are well known. The lines of centers of pressures for the normal loads and any variational loads to be expected or possible, are thus obtained for the assumed structure. And then, by modifying the assumed ellipse, the pressure center lines may be made to coincide with the section centers of the structure, and to fall at the center of the center third of the sections.

The nal structure as thus determined, may be adapted to, and be stable under, very wide variations of loading. In fact I have found that a stable partial elliptic structure may be designed so as to be stable under loads that vary from a minimum of substantially no external superimposed load at all, to a maximum load as high as any which can be expected to be ordinarily encountered. Thus, a structure may be designed which is perfectly safe and stable throughout a variation from substantially no load up to the very high exceptional loads to which structures like sub-street conduits and tunnels may occa` sionally be subjected.

Generally speaking, for any of the most usually encountered stresses, the ellipse which forms the basis of the design will be one which has a relatively large major axis and a relatively small minor axis. In practical designs for actual constructions, so far designed by me, the axes-ratios have run from 0.088 to 0.170-from about one-twelfth to about one-sixth. These ratios will always be less than one half, i, e. a minor fraction. The internal height of the structure will usually and preferably be a small fraction of the vertical semi-major axis of the ellipse, say not over one-third. The point of cut-off-the location of the oor with reference to the ellipse, is determined by the area of clear water way desired and also by economic considerations. Generally, for economic considerations alone, the internal maximum Width of the structure is preferably less than its height, the most economic proportion being in the neighborhood of the width equalling about eight-tenths of the height. However, the proportion of height to width of the structure is in many instances controlled by external conditions such as the vertical and horizontal space dimensions available for the conduit, or the vertical distance between the designed low line (the internal bottom line) and the ground surface. Where no such external conditions limit the choice of section, the economic conditions need only to be considered and the most economic proportions adopted. But the height to width ratio may be varied very widely in either direction without any substantial loss in economy and without at all departing from the principles of my invention. The structure may be either relatively high and narrow or low and broad.

I may mention also that my partial elliptic structure may be used in multiple tunnel structures with the same relative economies as in single tunnel structures.

As will be readily recognized my invention pertains to a structure and a structural form rather than to the uses to which the structure may be put. The invention may be put to any use to which it is suitable. In the following claims I use the term tunnel to indicate and include any arch structure which is designed or intended to carry an external load of any character.

I claim:

1. In a tunnel or the like having a floor, an arch wall springing at both sides from a base on the oor and having a central upper crown, and having the form of the upper portion of an ellipse whose major axis is vertical, the height of the arch crown above the arch base being substantially not more than a minor fraction of the semi-major diameter of the ellipse; all so that the center line of internal pressures within the arch wall, due to its own weight and finite uniformly distributed superimposed loading, lies wholly within the center third of the transverse section of the arch wall.

2. In a tunnel or the like having a floor, an arch wall springing at both sides from a base on the oor and having a central upper crown, and having the form of the upper portion of an ellipse whose major axis is vertical and whose minor axis is a minor fraction of its major axis, the height of the arch crown above the arch base being substantially not more than a minor fraction of the semi-major diameter of the ellipse; all so that the center line of internal pressures within the arch wall, due to its own Weight and finite uniformly distributed superimposed loading, lies Wholly within the center third of the transverse section of the arch wall.

3. In a tunnel or the like having a floor, an arch wall springing at both sides from a base on the door and having a central upper crown, and having the form of the upper portion of an ellipse whose major axis is vertical, the height of the arch crown above the arch base being substantially not more than substantially one-third of the semi-major diameter of the ellipse; all so that the center line of internal pressures within the arch wall, due to its own weight and nite uniformly distributed superimposed loading, lies wholly within the center third of the transverse section of the arch wall.

4. In a tunnel or the like having a floor, an arch wall springing at both sides from a base on the iloor and having a central upper crown, and having the form of the upper portion of an ellipse whose major axis is vertical and whose minor axis is not more than substantially onesixth of its major axis, the height of the arch crown above the arch base being substantially not more than substantially one-third of the semi-major diameter of the ellipse; all so that the Center line of internal pressures within the arch wall, due t0 its own weight and nite uniformly distributed superimposed loading, lies Whollyl Within the center third of the transverse section of the arch Wall.

ERNEST H. WILCOX. 

